| L(s) = 1 | − 28.5·2-s − 17.8·3-s + 303.·4-s − 2.27e3·5-s + 510.·6-s + 4.44e3·7-s + 5.94e3·8-s − 1.93e4·9-s + 6.50e4·10-s + 8.30e4·11-s − 5.43e3·12-s + 1.87e5·13-s − 1.27e5·14-s + 4.07e4·15-s − 3.25e5·16-s + 6.67e5·17-s + 5.53e5·18-s + 4.11e5·19-s − 6.91e5·20-s − 7.95e4·21-s − 2.37e6·22-s − 7.75e5·23-s − 1.06e5·24-s + 3.23e6·25-s − 5.36e6·26-s + 6.98e5·27-s + 1.35e6·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 0.127·3-s + 0.593·4-s − 1.62·5-s + 0.160·6-s + 0.700·7-s + 0.513·8-s − 0.983·9-s + 2.05·10-s + 1.71·11-s − 0.0755·12-s + 1.82·13-s − 0.884·14-s + 0.207·15-s − 1.24·16-s + 1.93·17-s + 1.24·18-s + 0.724·19-s − 0.966·20-s − 0.0892·21-s − 2.15·22-s − 0.577·23-s − 0.0654·24-s + 1.65·25-s − 2.30·26-s + 0.252·27-s + 0.415·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.086630069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086630069\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 1.50e9T \) |
| good | 2 | \( 1 + 28.5T + 512T^{2} \) |
| 3 | \( 1 + 17.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.27e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.30e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.87e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.67e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.11e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.75e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.43e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.79e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.14e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.73e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.02e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.23e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.31e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.56e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.86e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.24e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.28e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.05e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05485799354854337566997207586, −9.658112958623149784499351888057, −8.488549708611254426710376498850, −8.232093219561504759219735705175, −7.24026632112568866365621543623, −5.86570836986732187889687085464, −4.21819392318863602708663629787, −3.42444842162520505846402605313, −1.23941678130265676368181864720, −0.77043531220448362747622950414,
0.77043531220448362747622950414, 1.23941678130265676368181864720, 3.42444842162520505846402605313, 4.21819392318863602708663629787, 5.86570836986732187889687085464, 7.24026632112568866365621543623, 8.232093219561504759219735705175, 8.488549708611254426710376498850, 9.658112958623149784499351888057, 11.05485799354854337566997207586
